Restriction on minimum degree in the contractible sets problem

TL;DR

This paper investigates the existence of large contractible sets in 3-connected graphs, proving their existence under certain minimum degree conditions for sets of size at least 5.

Contribution

It establishes a minimum degree condition that guarantees large contractible sets in sufficiently large 3-connected graphs for sets of size at least 5.

Findings

Proves the existence of k-vertex contractible sets for k ≥ 5 under specific degree conditions.

Shows that a minimum degree of at least [(2k + 1)/3] + 2 suffices.

Supports the conjecture by McCuaig and Ota for large graphs with degree constraints.

Abstract

Let $G$ be a $3$-connected graph. A set $W \subset V(G)$ is called contractible if $G(W)$ is a connected graph and $G - W$ is a $2$-connected graph. In 1994, McCuaig and Ota conjectured that for any $k \in \mathbb{N}$ there exists $n \in \mathbb{N}$ such that any 3-connected graph $G$ with $v(G) \geqslant n$ has a $k$-vertex contractible set. It is proved that this holds if $k \geqslant 5$ and $\delta(G) \geqslant \left[ \frac{2k + 1}{3} \right] + 2$.